Mathematics, Decolonization and Censorship: C. K. Raju

Guest post by C.K.RAJU
Did you find math difficult in school? Does your child? If so, what is the solution: change the teacher or change the child? Blaming the teacher or the child for math difficulties is a common but unsound explanation. Thus, problems with teachers or students should equally affect all subjects, not only math.The right solution is to change math. That seems impossible. People naively believe that math is universal. In fact, the math taught today, from middle school onward, is called formal math; it began only in the 20th c. with David Hilbert and Bertrand Russell. It differs from the normal math which people earlier did for thousands of years, across the world, and still do in kindergarten.Formal math adds enormously to the difficulty of math but nothing to its practical value. The practical value of math comes from efficient techniques of calculation, used in normal math, not prolix formal proofs. For example, the proof of 1+1=2 took Whitehead and Russell 368 pages of dense symbolism in their Principia. That proof is a liability in a grocer’s shop. In contrast, normal math is easy. One apple and one apple make two apples as most people learn in kindergarten. So should we switch back to normal math at all levels?

Now our school texts justify the teaching of formal math as follows. The class 9 math text of NCERT1 (or of various states). tells the story of an early Greek called Euclid who was the first to do math “systematically” using deductive reasoning. The text further asserts that this was something that all others in Egypt, India, Iraq, China, and South America failed to do. It shows children an image of Euclid as a white man. On this story, students are told, we must do math by imitating “Euclid” who is glorified as the father of “real” math (meaning formal math).

The story condemns normal math as inferior. But no real argument is advanced to support the purported superiority of formal math, just a story. Do children check it? No; but the story is false. To expose its falsehood, I have offered a prize of Rs 2 lakhs for any serious (primary) evidence about Euclid. This prize stands unclaimed for several years. Why? Experts know2 there is nil evidence for “Euclid” and much counter-evidence.

Our own “experts”—the one’s who wrote the NCERT text—are unable to produce the evidence when challenged. They should either accept their mistakes, or defend their claims publicly, but do neither. Since we are totally dependent on such “experts” we just carry on with the wrong school texts! The vested interests involved are, however, deeper than just the vanity of “experts”, or their desire to preserve their jobs. Hence, attempts to publicly challenge the story of “Euclid” or to challenge the related philosophy of formal math are often censored.

As just one example of censorship, I wrote an article, “To decolonise math, stand up to its false history and bad philosophy”. This was published in the Conversation (global edition) in Oct 2016. The article created a stir. It went viral and recorded some 17K hits (60% in US and Africa) before it was abruptly removed by the South Africa editor. If there was something wrong in the article, the Conversation should have carried a public correction. No one was able to point to any actual error. So, the removal was privately justified on the lame editorial ground that I had “sited” (sic) my own work, such as my book.3 Even in India, the article was first reproduced and then taken down by both The Wire and Scroll, though to the credit of The Wire it put the article back with an apology. Currently, that censored article is available on my blog,4 on The Wire,5 and on Science2.0.6 It was also recently reproduced in full as part of another peer-reviewed journal article,7 so, again, there was nothing obviously wrong with it.

So, why was it censored? Why are false myths and censorship so essential to the teaching of math?

The answer involves three unpleasant facts. First, this way of teaching math came to us through colonial education,whichwas 100% church education when it first came to India in the 19th c.: not only mission schools, but all early Western universities such as Oxford and Cambridge were created by the church, and remained fully under its control till then.

Second, “Euclid’s” supposedly “superior” way of doing geometry was taught as part of the church curriculum for centuries. Why? That curriculum was designed to create missionaries. Future missionaries were taught the ability to persuade others: they were taught to use reason to persuade those who rejected the Christian scriptures. Hence, the church used math to teach reasoning, not practical calculation.

The third and least known fact is this: the word “reason” involves a tricky double speak. It does NOT refer to ordinary ways of reasoning, as people are easily tricked into believing. Rather it refers to a special way of reasoning developed by the church to support its “theology of reason” (which it adopted during the Crusades). Briefly, the church divorced reason from empirical facts. It had good reason to do so. Empirical facts are contrary to church dogmas: the notions of God, heaven, hell, resurrection, virgin birth are all contrary to the empirical. To defend its anti-empirical dogmas, the church declared empirical proofs to be inferior. It declared that “pure deductive proofs” based on reason, but divorced from facts, are infallible and “superior”. This church doctrine of reason is exactly what our school texts promote through the story of “Euclid” and his “superior” deductive proofs. Incidentally, that story also serves to hide the relation to church dogma.

Since the church used the book Elements as a textbook, its author had to be theologically correct, and early Greeks were the only people whom the church acknowledged as its “friends”. Hence, the author of the book was declared to be an unknown early Greek. The church never appointed a black or woman as pope, and it would have egg all over its face if it acknowledged the true author of the Elements as a heretical black woman who was raped and brutally killed in a church, as I asserted in my censored article. Science 2.0 did change the title of my article to “Was Euclid a black woman”, but did not add the part about her being lynched for being heretical.

The church used the book Elements by grossly “reinterpreting” the original. It was falsely asserted that the book contained pure deductive proofs, aligned to the church theology of using reason divorced from the empirical. This assertion, repeated by our school texts, is brazenly contrary to facts. The actual fact is that the book Elements does NOT have a single such pure deductive proof from its very first proposition to the last. Ironically, this itself shows how terribly fallible deductive proofs are—for centuries, invalid deductive proofs were wrongly accepted as valid by ALL Western scholars. When this truth was inevitably acknowledged, at the beginning of the 20th c., a quick substitute had to be invented to save Western pride from crumbling at a time when it was at its zenith.

The substitute for “Euclidean” math, invented by the West at the turn of the 20th c., was the formal mathematics of Hilbert and Russell. Russell’s proof of 1+1=2 is so complicated because one is not allowed to point out empirically that one apple and one apple make two apples. Formal mathematics mimics church dogma; it prohibits the use of the empirical, on the belief that the prohibition of the empirical leads to some “superior” form of truth.

This belief is pure balderdash. In fact, reason divorced from facts can be used to prove any nonsense whatsoever. To show this, I gave the example of the horned rabbit in my censored article. (1) All animals have two horns. (2) A rabbit is an animal, therefore, (3) a rabbit has two horns. Of course the conclusion is nonsense, and so is the premise (1). But we know that only as an empirical fact; if all reference to empirical facts is prohibited we have no way of knowing the truth or falsehood of premise (1). As Russell put it, in formal math we “take any hypothesis that seems amusing, and deduce its consequences”,8 and I am distinctly amused by the hypothesis that all animals have two horns, and its deduced consequences for rabbits. It illustrates the conclusions based on pure deduction which the church glorifies as infallible.

Others used reasoning differently together with empirical proof. For example, in India all traditional schools of philosophy accepted the empirical (pratyaksa) as the first means of proof. This was also true of traditional Indian math (normal math) from the time of the sulba sutra–s.9

Now, indoctrinated colonised minds often conflate acceptance of empirical with rejection of reasoning. But that is not true: like science, most systems of Indian philosophy, and traditional Indian math, accepted both empirical proofs and reasoning. The only exception was the Lokayata, or people’s philosophers, who warned against inference not based on the empirical. Their example of wolf’s paws is similar to the example of the horned rabbit above: on seeing the pug marks of a wolf, people in a city inferred that a wolf was around. Actually the pug marks were made by a man to demonstrate the foolishness of inference not based on sound empirical facts.

But this foolish dogma that avoiding empirical facts leads to a higher form of truth is what we still teach today. Early in middle school, children are introduced to formal math and avoidance of empirical as follows: the NCERT class 6 text asserts that a geometric point is invisible. It adds that a point determines a location. At two recent workshops, I asked a number of school math teachers and students how do they know what location a point determines since the point is invisible. They had no answer. But they had the honesty to admit their ignorance, unlike colonised intellectuals and “experts” who will defend the doctrine of invisible points exactly like the courtiers who defended the emperor’s invisible new clothes.

Many people say that math is difficult because it is abstract. This is wrong. The word dog is an abstraction, for dogs are of varying sizes and shapes. But children have no difficulty in understanding the abstraction “dog”, for one can easily point to a dog. Likewise children have no difficulty in understanding the abstraction dot, though dots come in various sizes and shapes and colours. But an invisible point is NOT such an abstraction: for one cannot point to a point. Nor can one infer the existence of invisible points from other phenomena the way one can infer the existence of electrons from tracks in a bubble chamber or infer fire from smoke. A geometric point, as taught in school today, is thus pure metaphysics; it has no real existence. People regrettably confound church metaphysics about unreal things with abstraction.

Further, a line too is asserted to be invisible by the NCERT 6th standard text. So I asked teachers and student how they can verify the postulate that exactly one straight line passes through any two points. They again had no answer. I also showed them that any two real dots can be connected by multiple straight-looking lines, so the postulate is not based on experience but solely on Western authority.

The church strategy of teaching about non-existent things forces students to abandon commonsense and rely on Western authority. Ultimately the only “reason” given by colonial education for why 1+1=2 is that some Western authority like Peano approved it! Those who resist this teaching, and try to understand on their own, are the one’s who find math difficult and abandon it.

But the text does not stop with one absurdity. After three years of allowing this nonsense about invisible points to sink in to the child’s mind, the NCERT class 9 text introduces a further piece of nonsense. It says that a point cannot also be defined in other words. (Ditto for line and plane.) It explains this as follows. If one says “A point is that which has no part” then one is obliged to define “part” and so on, leading to an infinite regress.

Such an infinite regress does NOT arise in the case of a dog or dot, for one can simply point to several instances of dots, terminating the regress. The cause of the infinite regress is the desire to preach the church dogma of avoiding the empirical. This motive is hidden, and never made clear by the NCERT text. Even the direct connection to church dogma is obscured by false stories of “Euclid”.

The matters of “Euclid” and the exclusion of the empirical are relatively simple. But if colonised minds have not understood even these simple tricks in two centuries, they are unlikely to ever understand the more complex tricks involved. The church would have permanently duped them through “education”. Thus, once students are conditioned to regard mathematics as pure metaphysics, a metaphysics of infinity crops up at every level: there an infinity of points in a line, an infinity of lines in a plane and so on. I reiterate that this metaphysics of infinity has nil practical value: a computer cannot handle any metaphysics of infinity, but most practical applications of math, such as sending a rocket to Mars, are accomplished using computers.

It is a common error to believe there is a unique notion of infinity, but I will not go into details here of how a particular metaphysics of infinity in math (especially calculus) forces us to accept church dogmas of eternity as part of science.10 Thus, the metaphysics which Europeans added to the Indian calculus forces time in physics to be like a line, as posited by core (post-Nicene) church dogma. This is the magic by which the metaphysics in math determines scientific “truth”. It is hard to explain this to people ignorant of math who are duped into thinking the belief in “laws of nature” is about science, though it is clearly a church dogma advanced by Aquinas in Summa Theologica.

We should change the teaching of math, and teach normal math solely for its practical value. Certainly we should do this at school level, for the benefit of the millions of students who drop out. They have no need to learn about the doctrine of invisible points and lines. The aim of colonial education was to teach people subordination to Western authority, but I dream of a new generation of children freed from the shackles which tie the colonised minds of many of our “educationists”.

But the tragedy is that the system cannot be changed. The students cannot change it. Even if the text is wrong, they are compelled to recite it under threat of failure. The teachers cannot change it, they must teach from the text or they risk losing their jobs. The government—ministers and bureaucrats—won’t risk changing it for they are ignorant and fear ridicule and loss of power if something goes wrong. The “experts” have a vested interest in it, so they will not change it. They refuse even to discuss things publicly.

There is an additional layer of protection which preserves myths and dogmas in math. This church method of censorship comes naturally to the colonised minds (intellectuals, journalists and so forth) who have been indoctrinated by the system. Unable to address an iota of the substantive critique, they will abuse, ridicule and reject and censor it. This censorship is done by loyal gatekeepers ignorant of both the philosophy of math and church theology. In this country of the blind, the two-eyed man is blinded because the colonially educated intelligentsia has been taught numerous subtle superstitions far more dangerous than astrology.

This is how the church controls mass behaviour—through mass superstitions—which it has smuggled into math and science to make those superstitions credible. Censorship defends that strategy. The only hope is that, today, people in the slums of Soweto understand what the colonised intellectuals elsewhere do not, that colonial education spreads superstitions packaged as part of math and science.

8Bertrand Russell, “Mathematics and the metaphysicians”, in Mysticism and logic and other essay, Longmans, Green and Co., London, 1919, p. 75.

9C. K. Raju, “Computers, Mathematics Education, and the Alternative Epistemology of the Calculus in the YuktiBhâsâ”, Philosophy East and West, 51(3), 2001, pp. 325–362. http://ckraju.net/papers/Hawaii.pdf.

C. K. Raju has authored several books, proposing a tilt in the arrow of time, and a new theory of gravitation in physics, and zeroism in math. He was the first to show that calculus developed in India and was transmitted to Europe in the 16th c. where it was misunderstood.

@Lefty Technocrat, This is the best you could get, is it? Raju’s piece may have a lot of polemical charge, but he is making an important argument. You reply with hot air? In case you have an argument – responding to the points raised in the piece, do try and put it together. And yes, don’t forget to use your own name.

Name calling is your only weapon he has presented an argument walrus you have no refutation so you rely on ad homynum attacks you lose buddy we Indians do not buy bullshit known as slander without proof

While it is an important argument and needs a lot more research and writing, as a social scientist I am deeply uncomfortable by the writer’s inclination to collapse complicated histories spanning several centuries and regions: Ancient Greece, Roman church, standardisation of education in Europe, British colonisation (which, let us not forget, does not follow the Catholic church). Many important arguments within postcolonial studies have been let down by the way evidence is presented. I am willing to be convinced but this article compresses histories and homogenises Europe in service of its argument.

@Rama. A lot of research has already been done, as anyone with even a little research background would have immediately noticed. Two books and several articles are cited in the post itself. Rama has not gone through them, for he raises no specific objection to even a single sentence in my 500 page tome Cultural Foundations of Mathematics or my book Euclid and Jesus. Had he seen at least the censored article, he would have found it cites a longer list of my books and papers posted at http://ckraju.net/papers/Reading-list-Bengaluru.html.

Does he contest my statements about Ptolemy in my booklet Is Science Western in Origin? Then there are my statements about Archimedes, also found in my recent lectures at the University of South Africa on “Not out of Greece”, posted at http://ckraju.net/unisa. Or my statement about Aristotle in my article on logic for the Springer encyclopedia? Then there are my numerous articles on decolonising education. Or is he challenging the physics which results from decolonized math? Say my theory of Lorentz covariant retarded gravitation which arises by debunking Newton’s and the formalist understanding of calculus and correcting the consequent error in Newtonian physics. BTW, the latest on that is at http://ckraju.net/papers/Flyby-experiment-article.pdf.

Rama passes a sweeping judgment without being able to contest even one sentence, in the thousands of pages of research I have published, so Rama’s opinion has no value. It is exactly the attitude of the censors: we don’t know, we don’t need to know, we reject because it doesn’t fit our beliefs.

Basically, he seems deeply uncomfortable that the grand myths of “Ancient Greece” with which he was indoctrinated have collapsed like a house of cards. He imagines they have more solidity. But, on my research, and as stated in the post, there is less than nil evidence for these myths. Hence, after a decade of attempts to debate my research, I offered a 2 lakh prize for serious (primary) evidence on Euclid. The prize is standing for seven years. Since Rama says I have not done adequate research he should quickly claim the prize. If he does not then his actions speak louder than his words.

Think of it. Is it ethical to teach these lousy church/racist/colonial myths to gullible school children if no expert in the world can produce evidence for them?

Rama makes a desperate attempt to suggest that Protestants were fundamentally different from Roman Catholics. But both are part of the Nicene creed, and both accept Augustine and also Aquinas’ theology of reason which came before the split. Both used education to breed missionaries. As for math, these myths were very much taught also in Protestant Britain; for example in Cambridge, where the norm was blind imitation of “Euclid” until 1887, after which some latitude was permitted, though not in “Euclid’s” definitions or common notions or the order of propositions. And there was no secular education in Britain until 1871, when it came only at the primary level, but “Euclid” continued to be taught for and in Cambridge.

As for convincing the indoctrinated, I have long maintained that it is harder to convince the colonised mind of its superstitions than astrologers. My key concern is to prevent indoctrination of the next generation. So, debate if you can, but ONLY on specific issues, or write a counter article and I will respond appropriately.

But don’t we have Archimedes palimpsests discovered in 20th century, which has all his works with proofs. Even his work on hydrostatics, quadrature of parabola, volume of paraboloid, spheroid, sphere etc. Also, without great tradition of geometry, how did the Greeks designed antikythera mechanism- the world’s first analog computer. Indians were nowhere close to Greeks in science and knowledge. Only later work of Kerala School can give some competition. Greeks had fundamentally more mature notion of science, while in India religion and spirituality was given more priority. Al Biruni in 10th century has written that Brahmins in spite of knowing science behind natural phenomena, preferred to lead masses to ignorance & superstition.

@Shashikant. Remarkably gullible. A palimpsest is a text which has been erased and overwritten by a religious text. There are two questions about the “Archimedes” palimpsest: (1) how do you know what was originally in it? (2) How do you relate it to Archimedes, considering that it comes some 1300 years after the purported date of Archimedes? As regards the first question, Heiberg lied about it, and it is impossible to see in it the things he fancifully described, as I have pointed out in my book Cultural Foundations of Mathematics. This is typical trick of church history: take an artefact and encourage all sorts of speculation about it, to spread the false story, as was done with the infamous “shroud of Turin”.

As regards (2) the connection of the palimpsest to “Archimedes” is achieved through another devious trick: the method of interwoven myths. The other myth, interwoven with the myth of the palimpsest, is that Archimedes is the author of Sphere and Cylinder. This second myth has no serious basis: as David Fowler admitted, that text comes to us from another land, in another language some 1800 years after Archimedes. But that 16th c. text reflects 16th c. knowledge. There is no serious way to relate that text to Archimedes. That trash story, that Archimedes was its author, is like saying a modern-day text on aerodynamics, in English, from London, is a verbatim copy of a 3rd c. Sanskrit text. And if you do trace the 16th c. text back, to Archimedes, why not trace it further back all the way to the Egyptians? After all, the Rhind papyrus offers unimpeachable PRIMARY evidence that Egyptians knew about the sphere and cylinder. That is far, far superior to any evidence for ANY claim of Greek achievements.

In summary, neither the palimpsest nor any text provides the slightest serious evidence for Archimedes or his works. But the stories are big, and some (“Eureka”) catch the imagination of children who are indoctrinated into them through our colonial education system, and a propagandist and ignorant press.

As for geometry, the Greeks only did religious geometry, for the soul, as described by Plato in his Meno, and Republic: a tradition continued until the 5th c. Proclus (who comments on the religious nature of the Elements of Geometry, wrongly attributed to “Euclid”). Specifically, Plato dismissed the practical applications of geometry. The Greeks were a superstitious lot and condemned people for doing science: e.g. the death sentence against Socrates was demanded on the grounds that he committed heresy by rejecting the divinity of the moon and saying it is a clod of earth. (http://ckraju.net/hps-aiu/extract-from-Plato-Apology.txt) Socrates denied the charge, saying he was not Anaxagoras who was similarly condemned. (See, Plato, Apology, and this extract. http://ckraju.net/hps-aiu/Plato-Apology-extract-2.txt). How could such a superstitious people do science? And indeed Greeks did no science. There is robust non-textual counter-evidence: the Greeks were arithmetically challenged like the Romans. (For details, see the video of my MIT talk. https://www.youtube.com/watch?v=IaodCGDjqzs) The Greeks had such a very poor calendar that Roman wits laughed at the Greek calends though the Julian calendar was itself grossly defective because Romans had no fractions. (Take a look at the video of my talk on the calendar. https://www.youtube.com/watch?v=MvpuC7Dg4e0) For more details, you can take a look at my UNISA lectures on “Not out of Greece” (http://ckraju.net/unisa/), or my easier book Is Science Western in Origin?

To compare with Indians, the first record of the experimental method, anywhere in the world, is in the Payasi sutta of the Digha Nikaya some 2000 years before Francis Bacon. The skeptic Payasi performed some 30 experiments to test the existence of the soul. Take a look at the first chapter in my book, The Eleven Pictures of Time. All schools of Indian philosophy (except Lokayata) accepted the empirically manifest, and inference as valid sources of knowledge (of any kind). A key issue here is that unlike the West, and even Popper, Indians did not exempt religious knowledge or math from the experimental method. Since, geometry was part of their religious knowledge, the West exempted math from the experimental method, resulting in formal math. This leads to religious beliefs creeping into present-day science, through formal math, as in the work of Hawking and Ellis. (See, the videos of my recent talks at the University of Cape Town (http://ckraju.net/blog/?p=135) to the great annoyance of Ellis and Co,)

As regards the antikythera how do you know the Greeks designed it? You are right that they could not have designed that instrument with such poor knowledge of arithmetic and astronomy. But they could always have copied it without understanding. (That is what Vasco da Gama did, when he first came to India with the help of an Indian navigator. He copied the instrument, called the kamal, without understanding it: he said that “the pilot [sic] was telling the distance by his teeth”, and took it back to have it “graduated in inches” (that is impossible, the instrument used a harmonic scale). If the Greeks could copy their gods and calendar and architecture from Egypt, why not navigational instruments? We know there is a very long tradition of astrolabes, going back to early Alexandria and Egypt.

Further, how exactly do you know the assigned date is valid? Because the artefact was buried under the sea for so long, there would all sorts of sediments attached to it, dating from various times, so it is very easy to fudge dates. Can you explain the exact details of what samples were dated? And how? Or did you just believe the reports published in newspapers, and start shooting based on that?

As regards, al Biruni, he is not an unbiased historian, but an emissary of Mahmood of Ghazni, a cruel despot, whom he dare not offend. So he does belittle Indians, especially to balance the fact that he is learning astronomy from India. Secondly, the particular superstition that al Biruni talks about is Rahu and Ketu. This has been picked up and repeated ad nauseum by numerous liberal historians like Romila Thapar. However, they are ill-informed. Lalla, two centuries earlier, devotes a whole chapter to “Elimination of false knowledge” (mithya-gyan-nirakaran). He specifically rejects the idea of Rahu and Ketu or other “artful demons” as the cause of eclipses. Both likewise reject the superstition that the earth is supported by someone (Greeks superstitiously believed the earth is supported by Atlas, Aryabhata denied it.) This is repeated by Vateswara in the 10th c. (For quotations see the originals or my chapter 4 in Cultural Foundations of Mathematics).

Can you point to any contemporary scholar in the Western tradition who dared refute any Christian superstitions? They very often encouraged them, For example, the date of creation in the Bible, computed with splendid accuracy by the Vice Chancellor of that church institution: Cambridge University. This is the point I make that colonial education is church education aiming to spread superstitions.

Likewise, Gallup polls have consistently shown that a stable 50% of the population in the US believes in the literal truth of church doctrines of resurrection, heaven and hell. So superstitions co-exist with scientific knowledge in any society. However, the curious aspect of Western society is that reputed “scientist” such as Stephen Hawking, G. F. R. Ellis, F. J. Tipler etc. (see video of my UCT panel talk https://youtu.be/ckbzKfRIi6Q) ENCOURAGE these superstitions using bad mathematics, and get millions of dollars as rewards. This is in the grand tradition of Kepler who earned a livelihood through astrology and called it divine harmony!

But the indoctrinated colonised mind cannot find any fault with the West: after all, we still use the inferior Christian calendar, which spreads superstitions. Ironically we use it to fix our secular festivals, and our identity. This is the whole art of Western propaganda: stuff the minds of impressionable and gullible children with the churchified history of Greek/Western origins of science, as all our school texts in math do. Children, like NCERT, need no evidence. They demand evidence only when the story changes.

A final point: you speak of the “Kerala school”, but this masks the fact that these highest caste namputiri Brahmins from the south, such as Nilakantha, regarded themselves as followers of Aryabhata, a lower caste person from Patna (http://ckraju.net/blog/?p=133). Acknowledging this, as also the pan-Indian development of calculus, presents a completely different picture of Indian society. It not only pushes the origin of calculus back to the 5th c., but shows its core to be a numerical method of solving differential equations, as I teach.

“Reason divorced from facts can be used to prove any nonsense whatsoever.” This is not really an argument against formal math because no one claims that formal math is going to produce conclusions that are true in the real world if you start with assumptions that aren’t.

Euclidean geometry derives its validity from being a good model for the geometry of our physical reality. I don’t think it matters much, as far as mathematics is concerned, whether there was a historical Euclid or if the Elements didn’t contain flawless proofs in the modern sense. Similarly, Peano’s axioms do not derive their authority from Peano or Dedekind, but from the fact that it is a good framework for doing arithmetic.

The notion of a point being “invisible” is not mathematically meaningful and is likely intended as a metaphor to aid the intuition that points and lines are “small” compared to the ambient space. There are however mathematically precise ways of saying something of this sort. If you are studying Euclidean geometry on the plane, which is two dimensional, points and lines are zero and one dimensional submanifolds respectively. Another example is that points and lines both have zero Lebesgue area measure. I am not justifying using the term “invisible”, but I do think that it is not entirely absurd as a metaphor for more difficult mathematical notions that cannot be introduced in sixth grade.

Rigorous mathematics is probably not of practical use to a grocer but does that make it unworthy of study? Literature and music are also not of practical use to a grocer either. The proof by contradiction that there are infinitely many primes(traditionally attributed to Euclid, but that is beside the point) is beautiful and worth thinking about for it’s own sake. I think it’s a mistake to value things only by how useful they are in your economic pursuits.

Russell and Whitehead’s proof of 1+1=2 or Hilbert’s formalism was not meant to make things as difficult as possible but an attempt come up with a set of universal axioms for all of mathematics. By the beginning of the 20th century, it was becoming increasingly clear that lack of rigor was leading mathematicians astray. It was in response to this and not for the purpose of upholding Western pride or church doctrine that mathematics was made more formal and rigorous.

When you make a statement that one apple plus one apple makes two apples, you are essentially using a definition of two as one plus one and therefore the statement is clear. That is perfectly fine and in fact most professional mathematicians don’t concern themselves much with formal logic either.

Claims such as the west is rationalist and the east empirical is minimizing both cultures.

Normal math, like science, accepts BOTH (a) facts and (b) reason. However, formal math rejects facts and accepts only reason. You are right that the theorems of mathematics are at best relative truths (relative to BOTH postulates and logic). However, my point is that there is no way to empirically verify the postulates of formal math since they all involve a metaphysics of infinity, even for 1+1=2 (there are an infinity of natural numbers). Likewise there are an infinity of points on a line, an infinity of lines in a plane and so on. So, I repeat the postulates are unverifiable, not merely unverified. Therefore, in math one never knows whether any proposition is true, as Russell accepted. But in normal math you know, as in science, whether it is at least approximately true. So, if math is done for its applications to science and engineering, then divorce from the empirical is good reason to reject formal math and accept normal math.

Two paragraph down you suggest that (formal) math should be done for aesthetics not (commerical) utility. This is an often-raised myth which I am tired of addressing. First, this claim naively conflates formal math with Platonic math (which talks of soul, and derives from Egyptian mystery geometry). The two have nothing in common: formal math is soul-less. Second, the claim that there is beauty in formal math involves an absurd notion of aesthetics, for the fact is that millions of students around the world drop out of math because they find it ugly and confusing. That is empirical proof that formal math is ugly. (Note, Plato, in Republic, recommended both music and math, but there is no need to teach students about aesthetics in music, which they love; formal math has turned math ugly.) Third, if it is about aesthetics, not utility, given the huge disagreement, have you taken the informed consent of parents and children before unethically imposing math as a compulsory subject in school? Fourth, as I argued in my Kosambi paper (http://ckraju.net/papers/Kosambi-EPW.pdf), you must openly inform parliament that math is being taught not for its utility but for some outlandish sense of aesthetics which most normal people reject. And you must take funds from the culture department not from well-funded science and engineering departments. I have no objection to teaching formal mathematics as an optional aesthetic pursuit at the post-graduate level, like Western music, if there are any takers.

Please desist from talking about “Euclidean” geometry: first produce evidence for “Euclid”. Second, delete the claim in the NCERT 9th std. text that the Greeks and especially Euclid, did math in some special way (using deductive proofs), for, contrary to the myth about it, the book Elements has no pure deductive proofs, from its first proposition to the last. Third, if the ultimate objective is approximate empirical knowledge there was nothing inferior about Egyptian or Indian cord geometry, which was normal math, so delete also those claims too in the NCERT 9th std. text, which declare that knowledge as inferior. Fourth, prove the fourth proposition in the Elements (SAS) empirically by superposition (as was originally done in the book), and remove all references to the SAS POSTULATE in current school texts, for there is nothing wrong in an empirical proof. Indeed, that is what I do in my Rajju Ganit text: this greatly simplifies the proofs of all propositions, including the “Pythagorean” proposition, which can be proved in one step. Lastly, recognize that there was no proper understanding of the “Pythagorean proposition” in the West since there was no square root algorithm, and this is required to calculate the diagonal (of a rectangle) from a knowledge of its sides as in Manava sulba sutra 10.10. Use the sulba sutra method, which accepts the square root of 2 as an approximation. See the cited paper on “Black thoughts matter” for more details.

As regards invisible geometric points, please read the NCERT 6th std. text. It is meant literally, and NOT metaphorically, just as surely the virgin birth of Jesus is meant literally and not metaphorically in post-Nicene Christianity (else Jesus is infected with original sin, and cannot save others from it). This literal meaning is exactly what you too assert when you speak of zero dimensions or zero Lebesgue measure. Note that the NCERT 9th std. text discusses similar possible definitions and correctly rejects them as invalid definitions, for they result in an infinite regress (to define dimension you must define a vector space, and likewise to speak of zero area, you must first define area; indeed, those definitions come AFTER the definition of point and are logically dependent on it). Your zero dimension or zero area points have to be invisible.

What the school text does NOT explain is that the infinite regress arises ONLY in formal math because of its strict anti-empiricism. There is no infinite regress in normal math in defining dots, which are visible, so that an ostensive definition is possible, as in my text. The idea that precision often increases when we reduce the size of dots is an idea that children easily understand. The need for mysterious invisible points arises only because of the mythical claim that (formal) math is EXACT.

What is rigorous math? As argued above formal math offers LESS certainty than normal math. So by “rigorous math” do you mean “less certain math”?

Your claim about rigor being the driving force is historically inaccurate. Please accept the fact that under overarching church influence, all Western scholars who studied the Elements blindly accepted the myths about it as valid for centuries. But eventually, after over five centuries, Western scholars started understanding what was patently obvious: that the proofs in the Elements (e.g. proposition 1, or SAS) were empirical proofs.

But the West did not reject the myth about the book; direct rejection would have been fatal to the authority of the church which used the book as a text to teach “reason” to its priests. They tried to “save the story”. They said the book was wrong, and it was not what the author intended to write! Laughable! They said they knew better the intentions of the mythical author! That is why Hilbert gives a “synthetic” interpretation of the Elements in his Foundations of Geometry. This is pure bunkum, for the interpretation does not fit the Elements. Beyond proposition 34: equality can no longer be interpreted as congruence, and Hilbert then then goes on to the absurdity of defining area while prohibiting length measurement. He did this because a metric interpretation (such as Birkhoff”s) trivializes the book and kills the story about something special in the particular arrangement of its propositions: the “Pythagorean” theorem can be proved in one step from Birkhoff’s metric axioms as was done empirically in India. So what you mean by “Euclidean” geometry: Plato’s geometry? Hilbert’s geometry? Birkhoff’s geometry? Or empirical compass-box geometry? Actually, our current school texts teach a hotch-potch and an incoherent mixture of axiomatic synthetic geometry and empirical metric (compass-box) geometry. That is what comes from blind imitation. The solution is to revert to pre-colonial Egyptian cord geometry or rajju ganit which is the best of the lot.

As for your claim that Dedekind cuts result in good arithmetic: this is not correct. First it is horribly messy: would you like to attempt a proof of 1+1=2 in REAL numbers from first principles, and write it out in full (including the needed formal set theory)? If it is too complicated to write down, it it too complicated to be taught in most elementary calculus courses. When it is taught, the accompanying formal set theory is not.

In normal math people used numbers like pi and sqrt(2) in a far easier way as “approximate numbers”. Further, Brahmagupta’s unexpressed fractions (rational functions) result in a far simpler (non-Archimedean) arithmetic which is better for calculus. I am NOT advocating it because that was the way calculus developed in India. I advocate it because calculus based on this arithmetic works BETTER than calculus based on formal reals or even the Schwartz derivative. See the appendix to my book on Cultural Foundations of Math.

I advocate no such caricature of Western reason vs Eastern empiricism. First of all, as I am tired of explaining, West refers to a state of mind, not a direction in space, so its opposite is non-West, not East. Second, my point is that post-Crusade the church adopted a theology of reason copied from the Islamic theology of reason (aql-i-kalam) the better to be able to convert Muslims and win the Crusades by other means. The church then spread the false superstition that anti-empirical reasoning is more reliable than reasoning based on the empirical. This was because the slightest contact with the empirical would destroy church dogmas. Hence formal math is based on the idea of accepting reason but rejecting facts. In India, as is well known, almost all schools of philosophy (except Lokayata) accepted the empirically manifest (facts) AND inference (reasoning), as both normal math and science do.

Lastly, I totally reject the idea that mathematics is about the practices of the formal mathematics community. Colonialism globalised colonial education and one particular way of doing mathematics. (As C. P. Snow points out, Cambridge University [a church institution] played a key role.) This created formal math community which wants to preserve itself and those beliefs and myths. Decolonisation means we critically examine the practices and beliefs of that community, and retain only what benefits the majority users of math instead of the “experts” or members of the community. However, it is a pity that, worse than astrologers, no “expert” or member of the community is willing to come forward and defend in public the formal math they teach.

@George Did you read this article, the author’s (counter) comments to the various comments here and equally important the references cited? Looks like you have not. Why I came to this conclusion.

The author’s point is with supporting evidence if you care to read this article, not like glossing over a newspaper article but like an academic paper, it will be clear the West is neither rational or empirical, it is Christian theological.

Also stop playing with words and confusing rigorous mathematics with formal mathematics. Don’t mean the same.

Clearly, you have no idea of depth of arguments or plain racist prejudiced in your views.